The cone of \( DD_n\)  (resp. \(SDD_n\) ) can be equivalently written as a number of linear constraints (resp. second order cone constraints), and thus linear optimization over the set of diagonally dominant matrices (resp.scaled diagonally dominant matrices) can be addressed using linear programming (LP) (resp. second order cone programming (SOCP) ) [1]}. Working with these classes of convex optimization problems allows one to take advantage of high-performance LP and SOCP solvers. However, as already pointed in [1]}, approximating the PSD cone by the set of scaled diagonally dominant matrices \( SDD_n\)  is conservative, which means the restricted problem may be infeasible or the optimal solution may be significantly different from that of the original SDP. Furthermore, enforcing these constraints could be problematic as \(n\)  increases, since the number of constraints grows in a combinatorial fashion \(\binom{n}{2}\) .
